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Mathematica > Mathematics and Algorithms > Discrete Mathematics > Combinatorial Functions >

Built-in Mathematica Symbol

CatalanNumber

CatalanNumber[n]
gives the n

Catalan number

The first 10 Catalan numbers:

Evaluate for large arguments:

CatalanNumber threads element-wise over lists:

Compute the number of different ways to parenthesize an expression:

Distribute over lists in AngleBracket:

Use the pattern matcher to repeatedly split the list into two parts in all possible ways:

The number of ways to parenthesize the expression a

Check:

The generating function for Catalan numbers:

The only odd Catalan numbers are those of the form CatalanNumber[2^k-1]:

Determinants of Hankel matrices made out of Catalan numbers:

Compare to Fibonacci numbers: